Bootstrap for testing the equality of selfsimilarity exponents across multivariate time series

Abstract

Because of the ever-increasing collections of multivariate data, multivariate selfsimilarity has become a widely used model for scale-free dynamics, with successful applications in numerous different fields. Multivariate selfsimilarity exponent estimation has therefore received considerable attention, with notably an original procedure recently proposed and based on the eigenvalues of the covariance random matrices of the wavelet coefficients at fixed scales. Expanding on preliminary work aiming to test for the equality of the selfsimilarity exponents in bivariate time series, we propose and study here a truly multivariate procedure that permits, from a single observation of multivariate time series, to test for the equality of several, possibly many, selfsimilarity exponents. It is based on an original bootstrap procedure, applied in a multivariate time-scale domain and designed to effectively capture the scale-dependent joint covariance structure of multivariate wavelet coefficients as well as the associated wavelet eigenvalue structures. Extensive simulations conducted on synthetic data, modeled by operator fractional Brownian motions, the reference multivariate selfsimilarity model, permit to show that the proposed multivariate timescale domain bootstrap based test yields the targeted significance level under the null hypothesis (all selfsimilarity exponents are equal) and to assess the power of the test for several alternative hypotheses. This analysis leads us to conclude that the proposed test for the equality of multivariate selfsimilarity exponents is effective and ready for use on (a single time series of) real data.

Counting the Number of Different Scaling Exponents in Multivariate Scale-Free Dynamics: Clustering by Bootstrap in the Wavelet Domain

“…In practice, the estimation of the selfsimilarity exponent is thus central and can be efficiently and robustly performed by means of wavelet transforms [8,9]. The multivariate time series encountered in many modern applications call both for multivariate selfsimilarity models, such as the recently proposed operator fractional Brownian motion (ofBm) [10][11][12][13], and for multivariate wavelet representation based estimation procedures, such as those developed in [2,3,14,15]. These estimation procedures output as many selfsimilarity parameter estimates as there are time series in the data.…”

“…The actual and relevant use of such collections of estimates for understanding the system under study requires, as a first step, determining how many of the estimates are actually different. In earlier works [15,16], we proposed a waveletdomain block-bootstrap strategy for deciding whether all selfsimilarity exponents are equal or not. Yet, this leaves untouched the critical issue of counting the number of equal selfsimilarity exponents.…”

“…To that end, the definition of multivariate selfsimilarity is briefly recalled in Section 2 together with the eigenwavelet-based selfsimilarity parameter estimation procedure. The proposed clustering procedure, the key contribution of the present work, is then detailed in Section 3: Building up on [15,16], the clustering strategy consists in an original multiple hypothesis test procedure for the equality of the reduced set of pairs of selfsimilarity exponents which, after ordering from small to large values, are neighbors. By construction, the so-obtained test decisions translate directly into clustering results.…”

“…computed at scale 2 j , provide relevant estimation of H. Yet, this approach was shown to suffer from the so-called repulsion effect in computing equal eigenvalues, thus inducing biased estimates for H [15]. To limit the resulting bias, it was proposed to compute modified wavelet spectra with equal number of wavelet coefficients contributing at each scale:…”

“…Following schemes proposed in earlier works [15,16], collections of bootstrap resamples of the multivariate wavelet coefficients D(2 j , k), k = 1, . .…”